yielding transition a dynamical perspective (energy ...zamponi/workshops/yielding...the local...
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Yielding transition A dynamical perspective
(energy landscape is not everything)
Jean-Louis BarratUniversité GrenobleAlpes
Institut universitaire deFranceInstitut LaueLangevin,TheoryGroup
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Outline
• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems• Creep
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Plasticresponse ofafoam (I.Cantat,O.Pitois,Phys.offluids2006)
Deformation ofamorphous systems atlow Tproceeds through wellidentified plasticevents orshear transformations(ArgonandKuo,1976)
Plasticresponse ofasimulatedLennard-Jones glass(Tanguy,Leonforte,JLB,EPJE2006)
Stress-straincurveatlowstrainrate,lowtemperature,smallsystems
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Eventsareshear transformationsofEshelby type
•Plasticinstability inavery localregion ofthemedium(irreversible)under theinfluenceofthelocalstress.
Malandro,Lacks,PRL1998•Instability involves typically afewtens ofparticles andsmall shear strains (1to10%)
•Surroundings respond essentially asanhomogeneous elastic medium(incompressible).Quadrupolarsymmetry oftheresponse. Puosi,Rottler,JLB,PRE2014
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Eventsareshear transformationsofEshelby type
Eshelby transformation:aninclusionwithin anelastic material undergoes aspontaneous changeofshape (eigenstrain):circular toelliptical.
Inanhomeogeneous,linear elastic solid,theInduced shear stressoutsidetheinclusionis proportional totheinclusiontransformationstrain andtotheEshelby propagator (response totwo forcedipoles):
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Eventsareshear transformationsofEshelby typeBestseen inexperiments trough correlation patterns
Colloidal paste under simpleshear(Jensen,Weitz,Spaepen,PRE2014)
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Bestseen inexperiments trough correlation patterns
Granular mediumunder uniaxial deformation(LeBouil,Amon,Crassous,PRL2014)
Eventsareshear transformationsofEshelby type
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• Microscopic :Particle based,molecular dynamicsorathermalquasistatic deformations.Detailedinformation,limited sizes /times.
• Mesoscopic :Coarse grainandusethe« sheartransformations »aselementary events,withelastic interactionsbetween them.
• Continuum :Stress,strain rate,andother statevariables(« effectivetemperature »)treated ascontinuumfields.
Three levels ofmodelling
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Mesoscopic description
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PhysRevA,1991
Springnetworkwiththresholdinforce
Slope-1.4in2D
Mesoscopic description- anold idea
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Outline
• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems• Creep
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Rheologyofsoftglassymaterials (SGR)By:Sollich,P;Lequeux,F;Hébraud,P;Cates MEPHYSICALREVIEWLETTERS Volume:78 Pages:4657-4660 Published:JUN161997
Very popular,based onBouchaud’s trap model
Mode-couplingtheoryforthepastyrheologyofsoftglassymaterials (HL)By:Hébraud,P;Lequeux,FPHYSICALREVIEWLETTERS Volume:81 Pages:2934-2937 Published:OCT51998
Less popular,probably much morerealistic
Two proposals fordescribing this scenarioinamean field manner
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13
The trap model (J-P. Bouchaud)
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•Exponential distributionofenergy barriers (->glasstransition)
• l strain variable,increaseslinearly with time
Avery popular model:SoftGlassy Rheology(Sollich,Lequeux,Hébraud,Sollich,Fielding)
P(L,E,t) distributionofsystems indifferent « traps »andatdifferent strains L.
Fixed strain rateevolution
Activated escapefrom traps dueto« mechanical noise »x
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Dynamical equation forthestrain distributionfunction P(E,l,t)onatypical site:
• Very successful model, describes many features of the flow o glassysystems + ageing
• glass transition at x=xg=1; power law fluid 1<x<2; Newtonian above• for x< xg : aging, yield stress sY, s=sY+A g1-x
But..Ømechanical temperature x is not defined self consistently, adjustableparameterØdoes it correspond to anything physical ?
Activated yield events
x=mechanical noisetemperatureResetstrain andenergyafter yield. G is thetotalplasticactivity.
External drive
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Thechallenger:Hébraud Lequeux model:Stressdiffusionduetomechanical noise+selfconsistency
⇧tP(⌅, t) = �G0�̇(t) ⇧�P +DHL(t) ⇧2�P � ⇤HL(⌅,⌅c)P + �(t) ⇥(⌅)
External drive Yield ifs>scStressdiffusion Resettozeroafter yield
P(s,t)probability distributionofstressonatypical site(nodisorder,singlelocalyield stress)
⇥HL(⇤,⇤c) ⇥1
⌅�(⇤ � ⇤c)
�(t) =1
⌅
Z
�0>�c
d⇤0 P(⇤0, t)DHL(t) = ��(t)
Nonlinear feedbackYield rule andplasticactivity
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Thechallenger:Hébraud Lequeux model:Stressdiffusionduetomechanical noise+selfconsistency
• � > �c = 2 Newtonian behaviour⇤ � ⇥̇
• � < �c = 2 Herschel Bulkley law with exponent 1/2:⇤ = ⇤Y +A⇥̇1/2
Maindifference between thetwo models:descriptionoftherandom process that triggerstheyield event.
• Solve for a fixed value of D (linear equation, P (�, D, �̇) is piecewise expo-nential).
• Obtain �(D, �̇) and enforce self consistency condition D = ↵�(D, �̇) )D(�̇)
• Obtain < � >=Rd��P (�, D(�̇), �̇)
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Mechanical noiseis different from thermalnoise!
A. Nicolas,K.Martens,JLB,EPL2014E.Agoritsas etal,EPJE2015
• Thermalnoiseactsonstrainvariablel inafixedlandscapebiasedbythestress
• Mechanicalnoiseactsadiffusiveprocessonthestressbiasitself
=>Verydifferentescapetimes(Arrheniusvsdiffusive)
PotentialEnergyLandscapePictureforasmallregion(STZ):
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Outline
• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems• Creep
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�̇ / (� � �yield)� “Secondorder”critical
behaviour,monotonousflowcurve.Avalanchebehavioratvanishingstrainrates,analogiesanddifferenceswithdepinningproblems.
Natureofthe«yield»(arrested->flow)transition?
Coexistenceofflowingandnonflowing regionsatthesamevalueofthestressisalsocommonlyobserved=>possibilityof“firstorder”transition,knownas“strainlocalisation”or“shearbanding”.”Spinodal”instabilityuponincreasingstrain->Procaccia etal.Herefocusonstationarystate,beyondyield.
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Strain localisation/Shear banding
Coexistenceofflowingregionsandsolidregionsatthesamevalueofthestress
Granular pastes(Barentin et al., 2003)
Bubble Rafts(Dennin et al., 2004)
Chocolate (Coussot et al.)
4.5 5.0 5.5 6.0
0.000
0.002
0.004
4.75 5.00 5.25 5.50
0.0
2.0x10-4
4.0x10-4
Velo
city
(m/s
)
Distance (cm)
Velo
city
(m/s
)
Distance (cm)
Lennard-Jones glass(Simulation, Varnik, Bocquet, JLB, 2004)
« Explained » by static vs dynamic yield stress
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Strain localisation/Shear banding
Divoux,Fardin,Manneville,Lerouge,Annualreviewfluidmechanics2016
Flowprofile(cylindricalCouette)
Flowcurve
Examplesystem
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Many different possiblemicroscopicmechanisms can leadtopermanentlocalisationofdeformation…Three examples here:longrecovery time(transient damage),inertia,friction
Strain localisation/Shear banding
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Strain localisation/Shear banding
Apossiblemechanismfromamesoscopic viewpoint(Coussot andOvarlez,Martensetal):longplasticevents(large“healingtime”)
Coussot andOvarlez meanfieldanalysis(EPJE2010)
Constitutivecurvebecomesnonmonotonicatlargetres
< ⇤ >= ⇥�̇ +⇤c
1 + �̇⌅res/�c
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Strain localisation/Shear banding
Apossiblemechanismfromamesoscopic viewpoint(Coussot andOvarlez,Martensetal):longplasticevents(large“healingtime”)
Lifecycleofasingleblock Flowcurves
KMartens,L.Bocquet,JLB,SoftMatter2012
Assemblyofelastoplastic blocksinteractingviaelasticpropagator.Healingtimetres beforeelasticrecoveryvaries.
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Strain localisation/Shear banding
Apossiblemechanismfromamesoscopic viewpoint(Coussot andOvarlez,Martensetal):longplasticevents(large“healingtime”)
Cumulatedplasticactivity
Martens,.Bocquet,JLB,SoftMatter2012Tyukodi,Patinet,Roux,Vandembroucq 2016“softmodesinthedepinning transition”
Elasticpropagatorreplacedbyshortrangeinteraction
Whylinearstructure?
=0
Outsideanhomogeneousplasticband(softmodeoftheelasticpropagator)
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Outline
• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems• Creep
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Strain localisationininertial systems
(SalernoandRobbins2014,NicolasRottler BarratPRL2015,Karimi Barrat2016)
Backtoamicroscopicmodel
Lennard-Jones particles, 2d systemDamping ⇥, mass m, stress scale �0 = �/⇤2 (in 2d).
Quality factor: Q = ⌅damp/⌅vib
Overdamped: Q ⌧ 1 underdamped Q � 1
⇥damp = m/� ; ⇥vib =p
m/�0
mv̇i = ��vi + F (xi) + ⇥i(t)
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(SalernoandRobbins2014,NicolasRottler BarratPRL2015,Karimi Barrat2016)
Overdamped system, zero temperature:
�(�̇, T = 0) = �0(0.72 + 2pW )
with W = ⇥�̇/�0
Highertemperature(T=0.2)Stressdecreases
Zerotemperature
Strain localisationininertial systems
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(SalernoandRobbins2014,NicolasRottler BarratPRL2015,Karimi Barrat2016)
Underdamped systems,zerotemperature:nonmonotonic flowcurves!
Ei = �̇p
m/�0 = �̇⇥vib
Grains in 3dQ ' 0.1a
p⇥�0/�
Strain localisationininertial systems
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(SalernoandRobbins2014,NicolasRottler BarratPRL2015,Karimi Barrat2016)
Interpretation:inertialvibrationsata“bathtemperature”T=0actasafinitetemperature
=>DataatlargeQcanbeobtainedfromdataatsmallerQandhighertemperature.
�(Ei,Q, T0) = � (Ei, 1, TK(Q,Ei, T0))
Rateweakeningeffectcompensatedatlargestrainratesbystandardincreasewithstrainrate
Energy dissipation proportional to ��̇
TK � C|�̇|+ T0
Strain localisationininertial systems
Nonmonotonic flowcurve – Shear bands?
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Nonmonotonic flowcurve – Shear bands?=>Stability analysis ofhomogeneous flow(K.Martens,V.Venkatesh,work inprogress).Assumemonotonous constitutiverelation:
Strain localisationininertial systems
ForceBalance
Temperature diffusion
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Strain localisationininertial systems
Homogeneous flowbecomes linearly unstable ifthesystemis larger than acritical size
Below this length scale heat diffusionis too fast andtheshear bandsdonotpersist intime.
Confirmed quantitatively bylargescale molecular dynamics simulations.
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Outline
• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems(workinprogresswithK.Karimi)
• Creep
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Experiments byAmon,Crassousetal
A.Amon,Habilitationthesis
Strain localisationingranular systems
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Experiments byAmon,Crassousetal
Correlationangle53°
Failureanglecloseto60°
Why arethetwo anglesdifferent ?
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Experiments byAmon,Crassousetal
• Biaxial testofgranular medium• Decorrelation ofspeckle patterngives accesstolocalplasticactivity (near thesurface).
• Correlation maps ofplasticactivity reportedduring deformation
Strain localisationininertial systems
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Stressredistribution+Failure criterion
Shear transformationaligned with x,yaxisandlocated attheorigin generates changesinthethree stresscomponents:
Stresstensor in2d
Strain localisationingranular systems
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Stressredistribution
Strain localisationingranular systems
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Failure criterionStrain localisationingranular systems
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Changeinyield function inresponse tosheartransformation
Strain localisationingranular systems
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CorrelationsStrain localisationingranular systems
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Shear band?• Plasticactivity localized inside alinearregion
• Picturebandasalinear array ofsheartransformations
• Ifactivity homogeneous ontheline,stressredistributionis zero everywhere outside
• Proposed criterion forselecting thebandorientation:maximise-dfy inside thebanditself
Strain localisationingranular systems
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Shear bandorientation
Different from correlation angleunless frictionangleis zero!
Strain localisationingranular systems
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Macroscopic friction
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Finite element grid.Each elementhaselastoviscoplastic behavior,.
Localfailure criterion with MohrCoulombcriterion.Localcohesion cdrawn from anexponentialdistribution,uniform localfrictionanglef.
Numerical test
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Simpleshear loading triggersfirstyield event
Solve forlocaldisplacement andstresses(overdamped propagation)
Triggernewevents iflocalfailure criterion isreached
Numerical test
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Loading curve
Smallstrains:Transientfluctuations
Largestrains:stable bands
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Correlations inplasticactivity
Correlations intheshear banded regime
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Correlations inplasticactivity
Correlations atsmall strain
Sliding average inthestraininterval 0– yield strain
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Outline
• Elastoplasticmodels• Meanfieldtreatments:Hébraud Lequeux,SGR• Strainlocalisation vscontinuoustransitions• Strainlocalisation ininertialsystems• Strainlocalisation ingranularsystems• Creep
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Siegenbürger etal,PRL2012Creep inacolloidal glass
Strain response:different stresslevels,different waiting times Fluidization timebehaves asa
powerlaw ofthedistancetoyield stress
Divoux etal,SoftMatter 2011Carbopol microgel
Creep:Apply afixed stresss andmeasure thestrain g (t)
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Stresscontrolled versionofelastoplastic models (mean field version)ChenLiu,KirstenMartens,JLBarxiv:1705.06912
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Results qualitatively similar toexperiments;very strong dependence ontheinitialconditionfortheprobability distributionfunction
Sd:decreases when systemages
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Fluidization timefollows apowerlaw with thestatic yield stressasareference (can be identified with overshoot instressstrain curve).
Exponent is notuniversal,depends onsystemage.
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Acknowledgements
KirstenMartens
AlexandreNicolas
EzequielFerrero
ElisabethAgoritsas
LydéricBocquet
ChenLiuFrancescoPuosi
KamranKarimi
JörgRottler
VishwasVenkatesh